In this paper we consider quantum Hamiltonians with time-dependent interaction strengths, and following the recently formulated generalized Bethe ansatz framework [Pasnoori, Phys. Rev. B 112, L060409 (2025)], we show that constraints imposed by integrability take the same form as the renormalization group flow equations corresponding to the respective Hamiltonians with constant interaction strengths. As a concrete example, we consider the anisotropic time-dependent Kondo model characterized by the time-dependent interaction strengths $J_{\parallel }\left(t\right)$ and $J_{\perp }\left(t\right)$. We construct an exact solution to the time-dependent Schrodinger equation, and by applying appropriate boundary conditions to the fermion fields we obtain a set of matrix difference equations called the quantum Knizhnik-Zamolodchikov equations corresponding to the XXZ $R$-matrix. The consistency of these equations imposes constraints on the time-dependent interaction strengths $J_{\parallel }\left(t\right)$ and $J_{\perp }\left(t\right)$, such that the system is integrable. Remarkably, the resulting temporal trajectories of the couplings are shown to coincide exactly with the renormalization group flow trajectories of the static Kondo model, establishing a direct and universal correspondence between integrability and renormalization group flow in time-dependent quantum systems.

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• Kondo effect
• Spin dynamics
• Bethe ansatz
• Exact solutions for many-body systems